> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# IGPC: Information-Geometric Perfect Cuboid
## Information-Geometric Perfect Cuboid — Propositions, Conjectures and World Layers

Version: v0.1  
Nature: Foundational propositions / world model / conjectures / redefining the pre-arithmetic layer of the Perfect Cuboid  
Canonical source of the original: UTF-8 Markdown  
Relations: X4-DMSD, GERES, DMSD, Perfect Cuboid Research Line

---

## Abstract

The Perfect Cuboid is usually stated directly as an integer or rational-number problem:

Find positive integers:

$$
a,b,c,d_{ab},d_{ac},d_{bc},g
$$

such that:

$$
a^2+b^2=d_{ab}^2,
$$

$$
a^2+c^2=d_{ac}^2,
$$

$$
b^2+c^2=d_{bc}^2,
$$

$$
a^2+b^2+c^2=g^2.
$$

This paper proposes an earlier-layer question:

> Before any radix, integer lattice or rational-number representation is specified, does there exist a complete, self-consistent "information-geometric perfect-cuboid world"?

That is, first study:

$$
\boxed{
\text{Information Geometry}
}
$$

rather than studying directly:

$$
\boxed{
\text{Integer Realization}.
}
$$

The basic objects in this layer are not decimal integers, but:

- Length relations;
- Ratios;
- Directions;
- X structure;
- Angles;
- Center;
- Face diagonals;
- Body diagonal;
- Shared edges;
- Exact coupling;
- Closure relations.

An information-geometric world can be realized in different arithmetic domains:

$$
\boxed{
\mathcal G
\rightarrow
\mathbb R
\rightarrow
\mathbb A
\rightarrow
\mathbb Q
\rightarrow
\mathbb Z,
}
$$

where:

$$
\mathbb A
$$

denotes algebraic numbers.

This paper formally distinguishes:

$$
\boxed{
\text{Geometric Existence}
}
$$

from:

$$
\boxed{
\text{Arithmetic Realizability}.
}
$$

Under this definition, existence of an information-geometric perfect cuboid in the real world is not a conjecture, but a directly provable proposition: take any three positive real edge lengths, define three face diagonals and the body diagonal exactly, and obtain a fully self-consistent cuboid relation graph.

What is genuinely unresolved is not:

> "Can this geometric world exist?"

but:

> "Can this geometrically realizable information world lie entirely on a rational lattice?"

Since any complete rational realization can have its denominators cleared to give an integer realization, classical Perfect Cuboid existence can be reformulated as:

$$
\boxed{
\text{Information-Geometric World}
\quad
\text{Does it admit a complete }\mathbb Q\text{-realization?}
}
$$

The paper's purpose is not to replace the classical problem with new terminology, but to make clear to researchers:

$$
\boxed{
\text{World Structure}
\neq
\text{Coordinate Representation}
\neq
\text{Arithmetic Type}.
}
$$

---

# 1. Why Redefine the Problem

The traditional Perfect Cuboid problem starts directly from:

$$
\mathbb Z
$$

that arithmetic domain.

But beginning from geometry / a world model naturally reveals a different level:

$$
\boxed{
\text{Geometric relations first, numerical representation afterward.}
}
$$

For example, the angle relationship between a rectangle's diagonal and edges is intrinsically independent of:

- Decimal;
- Binary;
- Hexadecimal;
- Floating-point format;
- Integer representation.

It depends only on ratios and relationships.

---

# 2. Representation-Free Quantity

Suppose a geometric quantity is:

$$
q.
$$

Decimal:

$$
10,
$$

Binary:

$$
1010_2,
$$

Hexadecimal:

$$
A_{16}
$$

may simply be different encodings of the same quantity.

Thus:

$$
\boxed{
\text{quantity}
\neq
\text{digit string}.
}
$$

---

# 3. Radix-Invariant Geometry

Define a geometric relation:

$$
R(q_1,\ldots,q_n)=0.
$$

If, after a representation change:

$$
\varphi:
\text{encoding}_1
\rightarrow
\text{encoding}_2
$$

it preserves the same relationship among mathematical quantities,

then:

$$
R
$$

is called radix-invariant.

All four quadratic constraints of the Perfect Cuboid are radix-invariant.

---

# 4. Information-Geometric World

Define an information-geometric world:

$$
\boxed{
\mathcal G
=
(
\mathcal O,
\mathcal E,
\mathcal X,
\Theta,
\mathcal R,
\mathcal C
).
}
$$

where:

### $\mathcal O$

Objects:

- vertices；
- edges；
- face diagonals；
- body diagonal；
- center.

### $\mathcal E$

Incidence / adjacency structure.

### $\mathcal X$

Diagonal-X structures.

### $\Theta$

Angle relations.

### $\mathcal R$

Ratio / metric relations.

### $\mathcal C$

Closure / coupling constraints.

---

# 5. Cuboid Relation Graph

A cuboid can be represented as a relation graph:

$$
\Gamma_{\mathrm{cuboid}}.
$$

Basic quantities:

$$
a,b,c,
d_{ab},d_{ac},d_{bc},g.
$$

Constraint edges:

$$
a^2+b^2=d_{ab}^2,
$$

$$
a^2+c^2=d_{ac}^2,
$$

$$
b^2+c^2=d_{bc}^2,
$$

$$
d_{ab}^2+c^2=g^2.
$$

The final equation is equivalent to:

$$
a^2+b^2+c^2=g^2.
$$

---

# 6. Definition: Information-Geometric Perfect Cuboid

Define an **Information-Geometric Perfect Cuboid**:

$$
\boxed{
\mathrm{IGPC}
}
$$

as a cuboid information world in which:

1. Three pairs of opposite edges are exactly paired;
2. Three face X structures are exact;
3. Every local edge–diagonal angle relation is exact;
4. The three face worlds share the same global edges;
5. The body diagonal and the three-dimensional edge world close exactly;
6. There is no advance requirement that every quantity belong to $\mathbb Q$ or $\mathbb Z$.

---

# 7. The Meaning of "Perfect" in IGPC

Perfect in IGPC does not mean:

$$
\text{Every length is an integer}.
$$

Instead:

$$
\boxed{
\text{Every information-geometric relation channel is exactly self-consistent}.
}
$$

Thus:

$$
\text{Perfect Geometry}
\neq
\text{Perfect Arithmetic}.
$$

---

# 8. Definition: Arithmetic Realization

Given an arithmetic domain:

$$
K.
$$

if a map exists:

$$
\rho_K:
\mathcal G
\rightarrow
K
$$

mapping every metric quantity into $K$ while preserving every constraint exactly,

then:

$$
\mathcal G
$$

is said to have a $K$-realization.

---

# 9. Real Realization

If:

$$
K=\mathbb R,
$$

it is called:

$$
\mathbb R\text{-realization}.
$$

---

# 10. Algebraic Realization

If:

$$
K=\overline{\mathbb Q}\cap\mathbb R,
$$

it is called:

$$
\mathbb A\text{-realization}.
$$

---

# 11. Rational Realization

If:

$$
K=\mathbb Q,
$$

it is called:

$$
\mathbb Q\text{-realization}.
$$

---

# 12. Integer Realization

If:

$$
K=\mathbb Z,
$$

it is called:

$$
\mathbb Z\text{-realization}.
$$

---

# 13. World Layer Hierarchy

Thus:

$$
\boxed{
\mathcal G
\rightarrow
\mathcal G_{\mathbb R}
\rightarrow
\mathcal G_{\mathbb A}
\rightarrow
\mathcal G_{\mathbb Q}
\rightarrow
\mathcal G_{\mathbb Z}.
}
$$

Not every arrow automatically exists.

---

# 14. Proposition IGPC-P1: Real-Geometric Existence

For any:

$$
a,b,c\in\mathbb R_{>0},
$$

define:

$$
d_{ab}=\sqrt{a^2+b^2},
$$

$$
d_{ac}=\sqrt{a^2+c^2},
$$

$$
d_{bc}=\sqrt{b^2+c^2},
$$

$$
g=\sqrt{a^2+b^2+c^2}.
$$

Then:

$$
\boxed{
(a,b,c,d_{ab},d_{ac},d_{bc},g)
}
$$

forms an exact IGPC $\mathbb R$-realization.

---

# 15. P1 Proof

Since square roots exist in:

$$
\mathbb R_{\ge0}
$$

that domain,

we directly have:

$$
a^2+b^2=d_{ab}^2,
$$

$$
a^2+c^2=d_{ac}^2,
$$

$$
b^2+c^2=d_{bc}^2,
$$

$$
a^2+b^2+c^2=g^2.
$$

Every relation channel is exact.

Q.E.D.

---

# 16. Corollary: Continuum Many Real IGPC Worlds

Because:

$$
(a,b,c)\in\mathbb R_{>0}^3
$$

has continuum cardinality,

there are also continuum many real realizations of IGPC.

Thus:

$$
\boxed{
\mathrm{IGPC}(\mathbb R)
\neq
\varnothing
}
$$

and this is not an isolated phenomenon.

---

# 17. Scale Quotient

If:

$$
\lambda>0,
$$

then:

$$
(a,b,c)
\mapsto
(\lambda a,\lambda b,\lambda c)
$$

preserves all normalized geometry.

Thus:

$$
\boxed{
\mathcal G/\mathbb R_{>0}
}
$$

is the scale-free shape space.

---

# 18. Normalized Shape Dimension

A generic cuboid is determined by:

$$
(a,b,c)
$$

three positive real parameters.

After quotienting out common scale, the shape space is:

$$
\boxed{
2\text{-dimensional}.
}
$$

For example, one can take:

$$
x=\frac ba,
$$

$$
y=\frac ca.
$$

---

# 19. Example: Unit Cube

Take:

$$
a=b=c=1.
$$

Then:

$$
d_{ab}=d_{ac}=d_{bc}=\sqrt2,
$$

$$
g=\sqrt3.
$$

This is a simplest exact real IGPC world.

---

# 20. This Example Is Deliberately Not a Rational Perfect Cuboid

Because:

$$
\sqrt2,\sqrt3\notin\mathbb Q,
$$

the unit cube shows:

$$
\boxed{
\text{IGPC geometric perfection}
\not\Rightarrow
\mathbb Q\text{-realization}.
}
$$

This is the most important distinction between levels in this document.

---

# 21. Proposition IGPC-P2: Integer Edges Always Give an Algebraic World

If:

$$
a,b,c\in\mathbb Z_{>0},
$$

then:

$$
d_{ab},
d_{ac},
d_{bc},
g
$$

are all real algebraic numbers.

Every integer-edge triple therefore has:

$$
\boxed{
\mathbb A\text{-IGPC realization}.
}
$$

---

# 22. Proof

Every diagonal is the square root of a positive integer:

$$
\sqrt N.
$$

It satisfies the polynomial:

$$
x^2-N=0.
$$

so it is algebraic.

Q.E.D.

---

# 23. Arithmetic-Type Vector

For an IGPC world, define the arithmetic-type vector:

$$
\boxed{
\tau(W)
=
(
\tau(a),
\tau(b),
\tau(c),
\tau(d_{ab}),
\tau(d_{ac}),
\tau(d_{bc}),
\tau(g)
).
}
$$

where the type can be:

$$
\mathbb Z,
\mathbb Q,
\mathbb A\setminus\mathbb Q,
\mathbb R\setminus\mathbb A.
$$

---

# 24. G2 as Mixed-Arithmetic World

For example:

$$
(a,b,c)=(448,495,840).
$$

has:

$$
448^2+840^2=952^2,
$$

$$
495^2+840^2=975^2,
$$

$$
448^2+495^2+840^2=1073^2.
$$

but:

$$
448^2+495^2=445729.
$$

Thus:

$$
d_{ab}=\sqrt{445729}.
$$

And:

$$
445729=233\cdot1913
$$

is not a perfect square.

Thus:

$$
d_{ab}\in
\mathbb A\setminus\mathbb Q.
$$

---

# 25. The Meaning of G2

G2 shows:

$$
\boxed{
\text{Geometry can be highly closed,
while just one arithmetic-type channel remains unrationalized.}
}
$$

This is not a failure of geometry.

It is:

$$
\boxed{
\text{arithmetic-realization failure}.
}
$$

---

# 26. Definition: Arithmetic Closure Vector

One can define:

$$
C_A(W)
=
(c_{ab},c_{ac},c_{bc},c_g),
$$

where:

$$
c=1
$$

means that the corresponding diagonal lies in the target arithmetic domain.

For a rational/integer target,

G2 gives:

$$
\boxed{
C_A(W_{\mathrm{G2}})
=
(0,1,1,1).
}
$$

---

# 27. Proposition IGPC-P3: Rational-to-Integer Equivalence

If IGPC has a complete:

$$
\mathbb Q
$$

realization,

there exists a:

$$
\mathbb Z
$$

realization.

---

# 28. Proof

Let all quantities be:

$$
q_1,\ldots,q_7\in\mathbb Q_{>0}.
$$

Take a common denominator:

$$
L.
$$

Then:

$$
Lq_i\in\mathbb Z.
$$

Every defining equation is homogeneous of degree $2$, so common scaling preserves it.

This gives an integer realization.

Q.E.D.

---

# 29. Corollary: An Equivalent Perfect Cuboid Problem

Classical Perfect Cuboid existence is equivalent to:

$$
\boxed{
\exists
\text{ IGPC world with complete }\mathbb Q\text{-realization}.
}
$$

The central question is therefore not:

$$
\mathbb R\rightarrow\mathbb Z
$$

but:

$$
\boxed{
\mathbb R\rightarrow\mathbb Q.
}
$$

---

# 30. Classical Problem Reframed

Traditional question:

> Does an integer Perfect Cuboid exist?

IGPC formulation:

> Among continuum many real IGPC worlds, is there at least one world whose metric channels all rationalize simultaneously?

---

# 31. Definition: Rationalization Operator

Let:

$$
\mathfrak R
$$

be a conceptual operator,

acting as:

$$
\mathfrak R:
\mathcal G_{\mathbb R}
\rightarrow
\mathcal G_{\mathbb Q}
\cup
\{\bot\}.
$$

If the world cannot be fully rationalized:

$$
\mathfrak R(W)=\bot.
$$

---

# 32. The IGPC Version of the Perfect Cuboid Conjecture

Define:

## IGPC-Q Conjecture

There exists:

$$
W^\ast\in\mathrm{IGPC}(\mathbb R)
$$

such that:

$$
\boxed{
\mathfrak R(W^\ast)
\neq
\bot.
}
$$

Equivalent to classical rational/integer Perfect Cuboid existence.

---

# 33. This Conjecture Does Not Follow from Real Existence

Although:

$$
\mathrm{IGPC}(\mathbb R)
$$

is very large,

one cannot conclude:

$$
\mathrm{IGPC}(\mathbb Q)
\neq
\varnothing.
$$

Because density of:

$$
\mathbb Q^n
$$

within:

$$
\mathbb R^n
$$

does not mean

that a particular algebraic subvariety necessarily has a rational point.

---

# 34. Proposition IGPC-P4: Radix Invariance

If two encodings:

$$
E_1,E_2
$$

represent the same mathematical quantities,

IGPC truth status does not change with radix.

That is:

$$
\boxed{
\operatorname{IGPC}(E_1)
=
\operatorname{IGPC}(E_2).
}
$$

---

# 35. Proof Sketch

IGPC constraints depend only on:

- addition；
- multiplication；
- square；
- equality；
- ratio；
- incidence.

These are quantity-level relations.

Radix changes only digit representation.

Truth is therefore invariant.

---

# 36. Radix-Free Information Sequence

A sequence can therefore be defined:

$$
\boxed{
\mathcal S_{\mathrm{IG}}
=
(
\text{objects},
\text{relations},
\text{ratios},
\text{angles},
\text{couplings},
\text{closures}
).
}
$$

Not:

$$
\boxed{
\text{decimal digits}.
}
$$

---

# 37. Why Floating Point Initially "Sees a Different Answer"

If floating point is used only to approximate a real realization,

it observes:

$$
\mathcal G_{\mathbb R}.
$$

Traditional integer search observes:

$$
\mathcal G_{\mathbb Z}.
$$

The two therefore answer different questions.

---

# 38. The Correct Floating-Point Claim

A floating-point / numerical world can support:

$$
\boxed{
\text{real-geometric feasibility}.
}
$$

but cannot by itself support:

$$
\boxed{
\text{rational arithmetic realizability}.
}
$$

---

# 39. The Role of Floating Point

Reasonable uses:

- Finding real manifolds;
- Estimating local geometry;
- Finding coupling;
- visualization；
- basin exploration；
- seed generation.

It cannot serve as an:

$$
\mathbb Q/\mathbb Z
$$

exact gate.

---

# 40. The X Fingerprint Belongs to the World Layer

For a rectangular face:

$$
\Xi(a,b)
=
\left(
\frac{a^2-b^2}{a^2+b^2},
\frac{2ab}{a^2+b^2}
\right).
$$

This describes X crossing structure.

Its meaning precedes decimal encoding.

---

# 41. Angle Fingerprint

$$
\chi(a,b;d)
=
\left(
\frac ad,
\frac bd
\right).
$$

This is also a quantity-level relation.

Thus:

$$
\boxed{
\chi,\Xi
}
$$

belong to information geometry, not digit representation.

---

# 42. Four-Edge Joint

The X4 joint:

$$
J^{X4}
$$

can be understood as:

$$
\boxed{
\text{A radix-free geometric interface object}.
}
$$

It stores:

- edge roles；
- ratios；
- X angle；
- diagonal relation；
- orientation；
- coupling slots.

---

# 43. The World-Layer Version of PerfectClick

Define:

$$
\operatorname{IGClick}(W)=1
$$

if all information-geometric interfaces are exactly compatible.

This differs from:

$$
\operatorname{PerfectClick}_{\mathbb Z}
$$

that arithmetic predicate.

---

# 44. Two Layers of Click

### Geometric Click

$$
\boxed{
\operatorname{IGClick}=1.
}
$$

Means that the world structure is exact.

### Arithmetic Click

$$
\boxed{
\operatorname{AClick}_K=1.
}
$$

Means that all quantities simultaneously lie in the arithmetic domain:

$$
K.
$$

---

# 45. Classical Perfect Cuboid

Is precisely:

$$
\boxed{
\operatorname{IGClick}=1
}
$$

and:

$$
\boxed{
\operatorname{AClick}_{\mathbb Q}=1.
}
$$

---

# 46. World Precedes Arithmetic Type

The methodological order advocated by this paper:

$$
\boxed{
\text{World}
\rightarrow
\text{Geometry}
\rightarrow
\text{Arithmetic Type}
\rightarrow
\text{Encoding}.
}
$$

Not:

$$
\boxed{
\text{Digits}
\rightarrow
\text{Search}
\rightarrow
\text{Hope for Geometry}.
}
$$

---

# 47. Conjecture IGPC-C1: Rational Realization

There exists at least one information-geometric perfect cuboid:

$$
W
$$

such that:

$$
\boxed{
W
\text{ has complete }\mathbb Q\text{-realization}.
}
$$

This is the world-model version of the classical Perfect Cuboid existence conjecture.

---

# 48. Conjecture IGPC-C2: Primitive Multiplicity

If a primitive rational/integer IGPC realization exists,

are there infinitely many pairwise non-similar primitive realizations?

Formally:

$$
\boxed{
\#(
\mathrm{IGPC}(\mathbb Q)/
\mathbb Q_{>0}
)
=
\infty?
}
$$

This is an independent conjecture stronger than classical existence.

---

# 49. C2 Differs from Ordinary Scale Multiplicity

Once an integer solution exists:

$$
W,
$$

then:

$$
kW
$$

gives infinitely many integer solutions.

What is genuinely interesting is therefore whether:

$$
\boxed{
\text{non-similar primitive shape classes}
}
$$

are infinite in number.

---

# 50. Conjecture IGPC-C3: Sparse Rational Intersections

One possibility:

$$
\mathrm{IGPC}(\mathbb R)
$$

is continuous,

but:

$$
\mathrm{IGPC}(\mathbb Q)
$$

is extremely sparse.

Formally, one may have:

$$
\boxed{
\text{real-positive dimension}
\quad
\text{with sparse rational lattice intersections}.
}
$$

---

# 51. Conjecture IGPC-C4: No Low-Complexity Sequence Law

Even if primitive rational realizations are numerous,

their ordered list:

$$
W_1,W_2,\ldots
$$

need not have a low-complexity:

- Linear recurrence;
- Single polynomial parameter;
- simple monotone gap law；
- Decimal digit pattern.

But this does not mean that there is no higher-dimensional structure.

---

# 52. A More Precise Meaning of C4

"No pattern" should be reformulated as:

$$
\boxed{
\text{There may be no low-dimensional, low-description-complexity pattern visible to humans.}
}
$$

The underlying structure must still obey:

$$
\Gamma_{\mathrm{cuboid}}
$$

those constraints.

---

# 53. Information-Geometric Family

The more natural research object may not be:

$$
\{W_n\}_{n\in\mathbb N},
$$

but:

$$
\boxed{
\mathcal M_{\mathrm{IG}}
}
$$

a continuous / algebraic / topological world manifold.

Integer solutions are merely its arithmetic-lattice intersections.

---

# 54. Lattice-Intersection View

The classical integer problem:

$$
\boxed{
\mathcal M_{\mathrm{IG}}
\cap
\mathbb Q^N
}
$$

or:

$$
\mathcal M_{\mathrm{IG}}
\cap
\mathbb Z^N.
$$

Whether a solution exists can therefore be viewed as:

$$
\boxed{
\text{world manifold}
\cap
\text{arithmetic lattice}
\neq
\varnothing?
}
$$

---

# 55. The Role of X4-DMSD

X4-DMSD is not responsible for proving existence of real geometry.

That layer is too easy.

It is responsible for:

$$
\boxed{
\text{Compiling information geometry into arithmetic-realization constraints}.
}
$$

---

# 56. The Role of DMSD

DMSD is responsible for:

$$
\boxed{
\text{Studying lattice intersections across scales
without expanding all arithmetic candidates}.
}
$$

---

# 57. The Role of GERES

GERES can in turn be used for:

$$
\boxed{
\text{real / approximate world}
\rightarrow
\text{exact world reconstruction}
\rightarrow
\text{arithmetic classification}.
}
$$

---

# 58. Division of Responsibilities Between the Three Research Lines

### IGPC

Defines "what the world is".

### X4-DMSD

Studies how the world enters an arithmetic lattice.

### GERES

Studies how an approximate observation recovers an exact world.

---

# 59. Formal Research Pipeline

$$
\boxed{
\text{IGPC World}
\rightarrow
\text{Real Feasibility}
\rightarrow
\text{Exact X4 Semantics}
\rightarrow
\text{Arithmetic-Type Projection}
\rightarrow
\text{DMSD Multiscale Search}
\rightarrow
\text{CRT / Lattice Closure}
\rightarrow
\mathbb Q\text{-Realization?}
}
$$

---

# 60. If Rational Closure Succeeds

If:

$$
W_{\mathbb Q}
$$

exists,

clear denominators:

$$
\boxed{
W_{\mathbb Q}
\rightarrow
W_{\mathbb Z}.
}
$$

The classical integer witness is therefore merely the final representation normalization.

---

# 61. If Rational Closure Always Fails

The information-geometric world still exists.

It is simply that:

$$
\boxed{
\mathrm{IGPC}(\mathbb R)
\neq
\varnothing,
\qquad
\mathrm{IGPC}(\mathbb Q)
=
\varnothing.
}
$$

This would be a very clear arithmetic-nonrealizability result.

---

# 62. Two Types of Nonexistence Must Be Distinguished

### Geometric Nonexistence

$$
\mathrm{IGPC}(\mathbb R)=\varnothing.
$$

This is false under the current definition.

### Rational Nonexistence

$$
\mathrm{IGPC}(\mathbb Q)=\varnothing.
$$

This is what corresponds to the classical no-Perfect-Cuboid conclusion.

---

# 63. Two Types of Existence Must Also Be Distinguished

### Geometric Existence

Already given by P1's trivial constructive proof.

### Rational Existence

Still open.

They must not be conflated.

---

# 64. Why It Is Still Worth Writing "The Obvious"

Because without separating the levels, researchers can easily conflate:

$$
\boxed{
\text{geometric possibility}
}
$$

with:

$$
\boxed{
\text{integer existence}
}
$$

in the same sentence.

The value of this paper is not that P1 is difficult,

but:

$$
\boxed{
\text{Layering the ontology}.
}
$$

---

# 65. Different Worldviews

Traditional perspective:

> Find integers.

IGPC perspective:

> First identify a relational world, then ask which arithmetic domains can carry it.

---

# 66. Information Geometry First

This can be written as a research principle:

$$
\boxed{
\textbf{World First, Arithmetic Second, Encoding Last.}
}
$$

---

# 67. Proposition IGPC-P5: Encoding Non-Primacy

If two realizations:

$$
W_1,W_2
$$

differ only in numerical encoding,

but their quantity relations are isomorphic,

then:

$$
\boxed{
W_1
\cong_{\mathrm{IG}}
W_2.
}
$$

Encoding is therefore not part of world identity.

---

# 68. Information-Geometric Isomorphism

Define:

$$
W_1
\cong_{\mathrm{IG}}
W_2
$$

if a bijection exists preserving:

- incidence；
- edge roles；
- ratio；
- angle；
- X structure；
- closure relations.

---

# 69. Scale Isomorphism

Common scaling:

$$
W
\mapsto
\lambda W
$$

is the simplest information-geometric isomorphism.

---

# 70. Arithmetic Type Is Not Mere Encoding

Note that the difference between:

$$
\mathbb Q
$$

and:

$$
\mathbb R\setminus\mathbb Q
$$

is not a radix-representation artifact.

It is an arithmetic property of the quantity itself.

Thus:

$$
\boxed{
\text{radix-free}
\neq
\text{arithmetic-type-free}.
}
$$

---

# 71. This Point Is Crucial to This Paper

"No radix" must not be misinterpreted as:

> Rational / irrational is unimportant.

The correct meaning:

> Radix does not determine truth, but the arithmetic domain still determines realizability.

---

# 72. Floating Point Is Not an Arithmetic Domain

IEEE floating point is a finite representation system.

It is not:

$$
\mathbb R.
$$

A numerical witness should therefore be understood as:

$$
\boxed{
\text{evidence / approximation to a real realization},
}
$$

unless there is a separate exact reconstruction certificate.

---

# 73. Exact Real-Geometric Certificate

The strongest real layer should not store only floating-point coordinates.

It should store:

- exact formulas；
- symbolic radicals；
- interval certificates；
- defining constraints.

Only then:

$$
\mathcal G_{\mathbb R}
$$

does not depend on floating-point error.

---

# 74. Suggested Research Roadmap

### IGPC-R0

Real information-geometric semantics.

### IGPC-R1

Radix-free X / angle / ratio canonicalization.

### IGPC-R2

Arithmetic-type projection.

### IGPC-R3

Rationalization gates.

### IGPC-R4

X4-DMSD linkage.

### IGPC-R5

Proof-carrying arithmetic-realization runtime.

---

# 75. The Core Certificate of IGPC-R0

A real-geometric world certificate contains at least:

$$
(a,b,c)
$$

or, normalized:

$$
(1,x,y),
$$

and:

$$
d_{ab},
d_{ac},
d_{bc},
g
$$

with exact symbolic expressions.

---

# 76. Example Certificate

Take:

$$
(1,1,1).
$$

Output:

$$
d_{ab}=d_{ac}=d_{bc}=\sqrt2,
$$

$$
g=\sqrt3.
$$

The four equations are exact.

Thus:

$$
\boxed{
\operatorname{IGClick}=1.
}
$$

But:

$$
\operatorname{AClick}_{\mathbb Q}=0.
$$

---

# 77. Mixed Arithmetic Certificate

G2 outputs:

$$
\operatorname{IGClick}=1,
$$

and:

$$
C_A=(0,1,1,1).
$$

It is therefore an exact world with "one arithmetic channel not yet closed".

---

# 78. Perfect Rational Certificate

If, in the future, one finds:

$$
C_A=(1,1,1,1),
$$

and all edges are rational,

then:

$$
\boxed{
\operatorname{AClick}_{\mathbb Q}=1.
}
$$

Clearing denominators immediately gives an integer Perfect Cuboid.

---

# 79. Final Restatement of the Proposition

The classical question can therefore be rewritten as:

$$
\boxed{
\text{Does the information-geometric perfect-cuboid world admit a full rational realization?}
}
$$

Its Chinese formulation translates as:

> **Does the information-geometric perfect-cuboid world admit a complete rational realization?**

---

# 80. This Paper's Claim Boundary

This paper has established:

- A definition of radix-free information geometry;
- The IGPC world object;
- Real / algebraic / rational / integer realization layers;
- The real-geometric existence theorem;
- Rational-to-integer equivalence;
- Arithmetic-type vector;
- Separation of geometric click / arithmetic click;
- Rational-realization reformulation of the classical Perfect Cuboid;
- Primitive multiplicity and sparse-rational-intersection conjectures;
- Division of responsibilities among X4-DMSD / GERES / DMSD.

This paper has not proved:

- Existence of a rational Perfect Cuboid;
- Nonexistence of a rational Perfect Cuboid;
- Infinitely many primitive solution classes;
- That G2 can deform into a rational Perfect Cuboid;
- That a floating-point approximation implies rational realization.

---

# 81. Conclusion

The most important point of this paper is not:

$$
\boxed{
\mathrm{IGPC}(\mathbb R)\neq\varnothing.
}
$$

This is almost obvious.

What genuinely matters is dividing the research object into:

$$
\boxed{
\text{World}
}
$$

$$
\boxed{
\text{Realization}
}
$$

$$
\boxed{
\text{Arithmetic Type}
}
$$

$$
\boxed{
\text{Encoding}.
}
$$

A world can be geometrically perfect,

yet unable to rationalize fully in arithmetic terms.

Thus:

$$
\boxed{
\text{Geometric existence is not the open problem.}
}
$$

The genuinely open layer is:

$$
\boxed{
\text{Rational arithmetic realization.}
}
$$

From this perspective, the Perfect Cuboid is no longer merely:

> Find a set of very large integers.

Rather:

$$
\boxed{
\text{Within a radix-free information-geometric world,
find an arithmetic realization lying entirely in }\mathbb Q\text{.}
}
$$

This is the world the paper hopes to make visible to other researchers.

---

## Next Step

The next paper could be:

# IGPC-R1 — Radix-Free World Semantics and Arithmetic-Type Projection

Formally establish:

1. information-geometric graph schema;
2. world isomorphism;
3. scale quotient;
4. $\chi/\Xi$ as radix-free invariants;
5. arithmetic-type lattice;
6. algebraic-number realization;
7. rationalization operator;
8. exact mapping between IGPC and X4-DMSD;
9. G2 mixed-arithmetic world;
10. arithmetic-realization equivalence diagram of the classical Perfect Cuboid.